Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x )^{2}( 1 - 2 x + 4 x^{2} )( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$ |
$1 - 10 x + 49 x^{2} - 158 x^{3} + 368 x^{4} - 632 x^{5} + 784 x^{6} - 640 x^{7} + 256 x^{8}$ | |
Frobenius angles: | $0$, $0$, $\pm0.117169895439$, $\pm0.333333333333$, $\pm0.478661301576$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $18$ | $52164$ | $15645798$ | $3526286400$ | $998327967318$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $15$ | $61$ | $207$ | $925$ | $4035$ | $16333$ | $65343$ | $262573$ | $1050915$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ae $\times$ 1.4.ac $\times$ 2.4.ae_j and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey 2 $\times$ 2.4096.hm_zkj. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.e $\times$ 2.16.c_ap. The endomorphism algebra for each factor is: - 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.16.e : \(\Q(\sqrt{-3}) \).
- 2.16.c_ap : 4.0.4752.1.
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 1.64.q $\times$ 2.64.ae_eb. The endomorphism algebra for each factor is: - 1.64.aq : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.64.q : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.64.ae_eb : 4.0.4752.1.
Base change
This is a primitive isogeny class.